Throwing a sailor overboard was not formal punishment

Was simply throwing an annoying sailor overboard a recognized naval punishment?

No. It is described as an unofficial prank or hazing act on a stopped ship, distinct from keelhauling.

Formal historical punishment differed from informal shipboard roughhousing. Keelhauling is a separate practice, while simply throwing someone into the sea was not presented as a standard sentence. If such a dunking happened, the ship would be stopped and the target would usually have provoked shipmates. The joking exchange still draws a useful line between authorized discipline and an unofficial prank or hazing custom.